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Properties

Label 1-67-67.42-r1-0-0
Degree $1$
Conductor $67$
Sign $0.575 + 0.817i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 − 0.281i)2-s + (0.142 + 0.989i)3-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (0.415 + 0.909i)6-s + (0.959 − 0.281i)7-s + (0.654 − 0.755i)8-s + (−0.959 + 0.281i)9-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.654 + 0.755i)12-s + (0.654 + 0.755i)13-s + (0.841 − 0.540i)14-s + (−0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.142 + 0.989i)3-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (0.415 + 0.909i)6-s + (0.959 − 0.281i)7-s + (0.654 − 0.755i)8-s + (−0.959 + 0.281i)9-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.654 + 0.755i)12-s + (0.654 + 0.755i)13-s + (0.841 − 0.540i)14-s + (−0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.575 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.624466238 + 1.361845718i\)
\(L(\frac12)\) \(\approx\) \(2.624466238 + 1.361845718i\)
\(L(1)\) \(\approx\) \(1.897484076 + 0.5379151004i\)
\(L(1)\) \(\approx\) \(1.897484076 + 0.5379151004i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.66497995982642230197464679984, −30.59895231877993175370056983801, −29.75982131424083795433660613572, −28.58783066562052238108883545872, −27.24150709568494334062901862782, −25.39388157671222861427344349927, −24.8243161870695794281123227036, −23.67644902455681687261765034676, −23.315602425496030840708064663385, −21.40492888369471921496736836965, −20.60077721858942843262715885189, −19.40706084467432994942104132207, −17.92353953120438089367392568085, −16.68320713515284300398617205420, −15.43969533028238837204802026924, −14.11186024434657118006170363818, −13.1460690425472238776997664526, −12.11680171573390966104069194736, −11.18676482242380862068752602489, −8.33479405447868541342069464990, −7.90562003941510622457442167626, −6.060352734313800747400355035537, −4.99055327214433182037181722503, −3.15562613915902760570577663023, −1.31802301088961198104824779193, 2.31469078630132454379406641556, 3.874342292850450180291647769936, 4.75785185076631241571747042403, 6.456941425290019563424096592741, 8.06697377358423358525607782795, 10.18214091523556551334989706898, 10.90463806532234534468414184150, 11.99454926887024388875716428376, 13.87032452788056421085210492990, 14.78237421131526749308978216049, 15.4305635148672889558813271997, 16.87578174755826690431198471586, 18.63437539325378523051523698642, 20.00779464001574228659843372266, 21.00977498367210565715772081409, 21.75608385240777184547825623287, 23.063536394974309186512339049522, 23.62606581575607716714763359891, 25.42687249070370520554969185893, 26.37960245969741778606722703914, 27.67972817289641160349734780946, 28.5404043771676392606615206237, 30.28150733220721213078730070207, 30.766151434900097028129831230084, 31.81222648875060984552322372772

Graph of the $Z$-function along the critical line